The maximum part of the kinetic wind energy per time unit passing the area spanned by a wind turbine that can be utilized is defined as the wind power, P (see e.g., ) and is written as 2P=β12ρAu3, where β is the Betz limit, ρ is the density of the air, A is the swept area of the rotors, and u is the wind speed.

An analysis of the actual or theoretical wind power potential would involve an analysis of the time series of P. However, a more practical approach is needed as current technology only allows turbines to produce power at certain wind speed intervals. Power production starts when the wind exceeds a “cut-in” value uci. Subsequently, total wind power production, PwT, increases according to PwT=β12ρAu3-uci3 until u reaches the rated wind speed value ur. The rated wind speed denotes the transition limit where the turbine starts to produce at rated power. For higher wind speeds, u>ur, PwT is kept constant until u reaches a “cut-out” value, uco. Thereafter the production is terminated (PwT=0) abruptly with increasing wind. This abrupt termination of wind energy due to too high winds is a storm shutdown and is called “storm control”. In practice, the described production regime for u>ur is brought about by instantaneous pitching of turbine blades. This pitching allows a portion of the energy to pass through the blades without utilization and is done to shelter the turbines from the harsh drag force and to minimize the turbines' maintenance. Consequently, the maximum power outtake for a turbine occurs when ur≤u<uco and can be written as Pwmax=const=β12ρAur3-uci3. In practice, Pwmax is the installed capacity. In order for our results to be as general as possible in the rest of the paper, and since the turbine park at each site is only imaginary and of unknown capacity, we normalize the power calculations Pw=PwTPwmax. The technological characteristics of the turbines thus result in a transformation P→Pw, which can be written as 3Pw=0,u<uciu3-uci3ur3-uci3,uci≤u<ur,1,ur≤u<uco0,u≥uco, where u is the wind speed data, uci=4 m s-1 is the cut-in wind speed, ur=13 m s-1 is the rated wind speed, and uco=25 m -1 is the cut-out wind speed. These numbers are retrieved from the SWT-6.0-154 turbines used in Hywind, Scotland . The SWT-6.0-154 turbines were selected as they are the turbines used in the world's first commercially operating floating wind farm off the coast of Scotland.

Some statistical quantities are studied to reveal both wind speed and wind power characteristics: for the wind speed the Weibull mean (μ) and standard deviation (σ), together with the scale and shape parameters (a and b), are calculated (see upper panel of Fig.  for an example wind speed distribution).

After conversion of wind speed to wind power via Eq. (), the data obtain an entirely different distribution (see lower panel of Fig.  for an example of wind power distribution). Calculating the arithmetic mean and standard deviation will most likely not result in values representing the typical wind power output and the associated variability. Instead, we use the median (q2) and interquartile range (IQR) as a measure of the middle value and the spread in the data, respectively. Both q2 and IQR are independent of the data distribution, which makes them adequate choices to represent the statistical characteristics of the wind power data.

The mean wind speed values (μ) for the five sites demonstrate that the potential for wind power harvest is very good, with the mean ranging from 9.97 to 11.25 m s-1. regarded the wind speed at 90 m a.s.l. in the Norwegian Sea and the North Sea as “superb” and ranked it in the highest wind category (category 7) with the potential to produce more than 400 W m-2 of wind energy. By comparison, many of the wind parks already operating in the Yellow Sea (east of China) are only ranked in categories 4–6, ranging from “good” to “outstanding”, with the potential to produce 200–400 W m-2, respectively.

As mentioned earlier, the wind power intermittency is a huge problem due to the balancing difficulties and high economic costs related to a fluctuating power output. Among the five platforms, Ekofisk (ek) has the lowest variability, with a standard deviation of 5.3 m s-1. Gullfaks C (gf) is the site with the highest variability, where σ=6.0 m s-1. a is the scale parameter giving the height and width of the Weibull distribution. A large (small) scale parameter indicates a wide and low (high and narrow) distribution. For this data set a ranges from 11.2 to 12.7 for Draugen (dr) and Sleipner (sl), respectively. b tells us about the shape of the distribution. A small b (b<3) means that the distribution is positively skewed, with a long tail to the right of the mean. The smaller the number, the more right-skewed the distribution. Here, b ranges from 1.73 to 2.09 for Draugen (dr) and Ekofisk (ek), respectively, meaning that all the distributions are positively skewed, confirming that the wind speed has more of a Weibull distribution than a Gaussian distribution. Since the goal of a functioning wind park is to produce as much wind power as possible, it is desirable that the wind speed data fall between 4 m s-1≤u<25 m s-1, or even more preferably between 13 m s-1≤u<25 m s-1 which will give Pw=1. Sleipner (sl) is the site that most frequently operates at rated power: slightly over 30 % of the time. This is a result of the fact that Sleipner (sl) has an optimal combination of the scale (12.3) and shape (1.9) parameter, with the largest portion of the wind speed distribution falling between 4 and 25 m s-1.

For the wind power, the median values (q50) range from 0.33 for Draugen (dr) to 0.52 for Norne (no). Another measure of the performance of a wind park is the capacity factor (CF), which is defined as the annual mean power production divided by the installed capacity. Draugen (dr) has the lowest capacity factor, CF=0.45, and Norne (no) has the highest, with CF=0.53. The wind power IQR is high, around 0.9. This means that q50±IQR2 contains 50 % of the wind power output. There is therefore potential for reducing wind power intermittency by combining sites.

concluded that the coefficient of variability for wind power can be substantially reduced by geographic allocation of the production sites. They used the arithmetic mean (μ) and standard deviation (σ), calculating the coefficient of variability (CV=σμ). As mentioned above, using the arithmetic mean and standard deviation gives rise to a misleading interpretation of the actual middle value and the accompanying variability of the wind power output. This is due to the very different wind power distribution arising from the nonlinear conversion of the wind speed data to wind power output through the power conversion curve (see Eq. ). Hence, another more robust and resistant measure of wind power variability is the RCoV . The RCoV=MADq50 is the median absolute deviation (MAD) divided by the median (q50) and is a normalized measure of the spread in the data set. Here, RCoV ranges from 0.9 for Ekofisk (ek) to 1.00 for Draugen (dr), meaning that a typical deviation from the median value is approximately equal to the median value itself.

In terms of reducing wind power variability, the optimal hypothetical case would have been to combine production from two stations with correlation coefficient r=-1. The two combined sites would then be completely out of phase, and the sum of the individual productions would be constant in time, given equally installed capacity at each site. Since the synoptic weather systems constitute the main source of spatiotemporal variance in wind over open ocean, the connected production sites situated far apart in our site array will probably experience the most contrasting weather and therefore result in the largest dampening of the wind power intermittency. To quantify this, we investigated the pairwise correlation between the different wind power time series (Pw) as a function of distance and time lag between the hypothetically connected sites. analyzed the characteristics of the wind power resource for 66 onshore sites in the United Kingdom. He found a substantial reduction in the correlation between site pairs at separation distances less than 600 km. However, for wind sites located more than 800 km apart, the decrease in correlation is less pronounced, despite a further increase in the distance between them. Our results confirm this distance dependency of the correlation found in : Fig.  illustrates how the correlation between station pairs changes as a function of the separation distance. The correlation drops off quickly as the distance (x) between the sites increases. After x≈800 km the decrease in correlation with distance is reduced to 0.1 and continues to decrease more slowly with increasing separation distance. It is expected that the correlation between site pairs will approach zero when the separation distance becomes large enough, meaning that the wind at these sites is completely independent. Some site pairs can even have slightly negative correlation. Reaching this slowdown in the relation between the correlation and the separation distance after x≈800 km indicates that combining sites outside a radius of x≈800 km for further variability reduction has almost a negligible effect for the length and timescales considered here. Nevertheless, the correlation coefficient never drops to zero, or below zero, over the range of the data covered in this study, indicating that none of these station pairs will either be anticorrelated or have completely independent production (r≤0).

The decorrelation length illustrates at what radius the wind power correlation drops to a fraction of the initial value at x=0. In our study, we use the e-folding distance

The distance where the correlation has dropped to 1e=0.37.

To study the effect of interconnected production sites, we have to make combined wind power time series for all the different site configurations. The collective wind power time series, Pwc, for different configurations of sites are found by calculating the average power production for each time step. Time steps containing NaN values for any of the sites in the configuration in question are not included. Hence, for a given time step, 4Pwc(i)=1j∑j=1JPwj(i), where i=1, 2, …, N is the time step, j=1, 2, …, J is the number of sites combined, and Pwj(i) is the different wind power time series for an array combination of j sites (one to five sites) at time step i.

For example, a combination of two sites (a and b) at time step i will be 5Pwc(i)=12∑j=12Pw2(i)=12Pwa(i)+Pwb(i). This calculation is done for each time step i, as long as Pia≠NaN or Pib≠NaN.

As mentioned in Sect.  the IQR is a candidate for estimating wind power variability . According to a more robust and resistant variability measure is the robust coefficient of variability (RCoV). Figure  presents IQR and RCoV for different configurations of collective wind power production, Pwc. Common to both the variability measures is that the variability generally decreases quickly with increasing array size of connected sites. This result clearly demonstrates the advantage of having interconnected wind power production in terms of intermittency reduction: Instead of operating wind turbines at two sites separately, we see that the intermittency of a connected site pair is reduced and is further reduced with increased array size. The site with the highest (lowest) IQR is the station with the lowest (highest) RCoV, since RCoV is normalized by the median value.

A counterintuitive result is that for some site combinations the variability (IQR and RCoV) is less than the variability for a larger array size. For example, the combination of ek + sl + gf has a higher variability than most of the pairwise site combinations. Some pairwise combinations even have a lower IQR and RCoV than a four-site combination. This result appears to be a consequence of the geographic locations of the sites in this study. Since Ekofisk (ek), Sleipner (sl), and Gullfaks C (gf) are roughly aligned in a north–south direction, they will experience the same wind event more or less simultaneously (see Fig. ) due to the passage of extratropical cyclones and the associated fronts. Therefore, a combination of these sites would be poorer in terms of intermittency reduction than other site combinations, and even combinations of smaller array size.

The long record of observations (16 years) enables us to make estimates of the risk of having critically low wind power production. In this study, we have chosen to examine the time fraction the wind power production is zero (Pw=0 or Pwc=0).

The risk (Ri) of zero wind power is the sum of the risk of having too low (Rilow) and too high (Rihigh) wind speed, Ri=Rilow+Rihigh. Ri is calculated by taking the sum of all the hours the power is zero divided by the total number of time steps (NaN values are not included) and is calculated in the following way: 6Ri=1n∑i=1niδ,δ=1ifu<4∧u≥250else, where i is the time step and δ is a modifier and takes on the values 0 or 1, depending on the wind speed u. This formula is valid for both the individual wind power time series and the collective wind power time series (see Sect. ).

A critical question, given a pairwise connection of sites, is how much wind power one of the site produces when the other site is not producing any at all. Figure  presents the median wind power production at one site when the wind speed at the other station is causing zero power production. As the distance between two connected sites increases, the median production at the producing site increases. To ensure a median wind power production of at least 25 % of installed capacity at one site when the connected site is not producing any power at all, the separation distance needs to exceed ≈600 km for the investigated region.

Since wind is a fluctuating physical parameter, the risk of having a wind speed less than a given low threshold value is rather high. In addition to the well-known smoothing effect achieved by geographic allocation of wind power, also investigated the resulting impact on low-power events. They found that wind power production below 15 % of installed capacity was hardly ever observed when sites were combined. In contrast to , we study the effect of interconnecting sites on unwanted zero events. However, the effect is similar to that found by , namely that the occurrence of unwanted events is reduced. The left panel of Fig.  illustrates how the risk of zero events changes with distance between the sites and with array size of connected sites. Going from a single production site to two sites, the risk of zero power drops dramatically: from a risk of 8 %–11 % for a single site to a risk of less than 4 % for two connected sites. Note that the site combination of Draugen (dr) and Norne (no) has a small risk of Pwc=0, despite the short distance between them (160 km). The reason is probably the high wind power production at Norne (no), caused by the topographic effect arising from wind interactions with the topography in Norway. Hence, the high wind power at Norne (no) compensates for the relatively low wind power production at Draugen (dr) (see Table  for median Pw and CF values). This is relevant for cost estimates related to interconnection. This result also underlines the need for careful selection when connecting neighboring sites in terms of intermittency reduction. also found that by connecting four offshore wind farms the occurrence of zero events was reduced from 9 % to 4 %. By contrast, when we connect four of our sites, the risk of having Pw=0 is less than 0.5 %. The significant reduction of the risk in this study is due to the greater separation distance between the sites.

An even more detailed view of the risk of having Pwc=0 can be seen in the right panel of Fig. . This figure tells us which site configuration that has the lowest risk of having Pwc=0. As can be seen in the upper panel, the largest risk reduction is achieved when shifting from individual site production to a combined two-site production. Increasing the array size further reduces the risk, but the reduction is smaller. The configurations with an array size of three or more have a risk of less than 0.5 % (except the combination of Ekofisk (ek), Sleipner (sl), and Gullfaks C (gf) which has a risk of ≈1 %). This indicates that increasing the array size beyond three might not be financially sound. The fact that the intermittency reduction ceases is in accordance with the result obtained by . They found that at a frequency of 1 h-1 the high- to low-frequency variability was reduced by 87 % when combining four sites, compared to a single production site, and that increasing the array size by the remaining 16 sites resulted in a further intermittency reduction of only 8 %.

Since a zero event (Pw=0) occurs when both u<4 m s-1 (too low wind) and u≥25 m s-1 (too high wind), we choose to split the zero-power events, when investigating associated meteorological processes. Figure  presents both the occurrence and duration of zero events for two seasons, namely winter and summer, when the most contrasting results occur.

The first thing to notice is that the occurrence of zero events decreases as the duration increases. In addition, the occurrence of zero events has almost ceased when all the sites are connected (black curve), and the reduction is most distinct for the shortest duration (1–3 h). More zero events are caused by too low wind (a total of 684.5 yearly zero events for all the sites) than too high wind (102.75). The occurrence of zero events caused by too high wind (too low wind) is highest during winter (summer). In addition, we see that the seasonal difference is largest for the zero events resulting from too high winds, where the occurrence during the winter is of a larger magnitude than during the summer.

To explain the seasonal differences in the occurrence of zero events, it is necessary to examine the main driving force of variability in the weather phenomena over the open ocean. Synoptic high- and low-pressure systems give rise to the changing weather at our specific sites and are the main contributor to weather variability over the open ocean. Winds strong enough to terminate power production (u≥25 m s-1) are often associated with the passage of intense low-pressure systems and their accompanying fronts. The occurrence of such strong wind events is more likely to take place during winter than during summer at the latitudes in question . Throughout the year, differences in solar insolation give rise to an increased meridional temperature gradient during the northern hemispheric winter. These winter conditions result in a stronger background flow that favors low-pressure activity. On the other hand, the lack of these strong low-pressure systems during summer is probably the main reason why the occurrence of zero events caused by too low wind is highest during summer. In addition, the fact that blocking high-pressure events are more likely during spring also contributes to the seasonal difference in the too-low-wind events .

The previous section demonstrated that the occurrence and duration of zero events are sensitive to the season and the atmospheric state. has related long-lasting above- or below-average mean monthly values to variability in selected large-scale atmospheric circulation patterns. This section more closely examines surface pressure patterns associated with zero events lasting longer than 12 h using NORA10-reanalysis data .

Surface pressure is a key quantity that contains considerable information about the atmospheric structure in the lower atmosphere. Figures  and  present the average surface pressure conditions (composite mean) and the corresponding standard deviation associated with zero events due to too low wind (u<4 m s-1) and too strong wind (u≥25 m s-1) for the two site combinations ek + sl and dr + no, respectively. The average atmospheric condition (comprised of 360 maps) resulting in too low wind speed for ek + sl is a high-pressure system extending from the North Sea and into the Norwegian Sea. This pattern is similar to the positive phase of the Scandinavia pattern (SCAND) . The variability given by the standard deviation (SD) is only a few hectopascals, indicating that the atmospheric patterns causing too-low-wind events are relatively similar to each other and that most of the variability lies in the patterns' extension towards the west. By contrast, the average atmospheric situation (comprised of 15 maps) due to too strong wind is an intense low-pressure system hitting the northwest coast of southern Norway, bringing tight isobars and strong winds over Sleipner (sl) and Ekofisk (ek). This situation is similar to the positive phase of the North Atlantic Oscillation (NAO). The SD is large in the center of the low-pressure system. The small spatial extension of the maximum standard deviation indicates uncertainty regarding the depth of the system. However, the SD is less over Ekofisk (ek) and Sleipner (sl), indicating that these sites seem to be located to the south of the extratropical cyclone where the strongest winds are often found.

Figure  contains the same information as in Fig. , but for the site combination Draugen + Norne (dr + no). The average atmospheric condition (comprised of 159 maps) caused by too low wind is different from that in Fig. a. The typical atmospheric condition here is a high-pressure system covering the entire Norwegian Sea and extending across Norway and into eastern Europe. The SD states that the eastern extension of the high-pressure system is unclear, giving rise to the large uncertainty east of Norway. Again, as in the case of too high wind for ek + sl, the site combination dr + no is located to the south of a strong low-pressure system (panel b). For dr + no, the mean system is now situated off the coast of northern Norway. The corresponding SD is large, indicating that several positions and strengths of the strong low-pressure system can cause situations with too high winds for dr + no.

Even though investigated only four specific meteorological situations giving high and low collective wind power, our results are in line with theirs. As for the too-low-wind events in this study, the episodes of too low wind power in were associated with a high-pressure system located in the vicinity of the wind power production sites. On the other hand, the episodes resulting in high collective wind power output in were characterized by a low-pressure system located in the vicinity of the production sites. This is more or less in line with the results of this paper: our too-high-wind events are caused by intense low-pressure systems. Accordingly, a less intense low-pressure system will result in high-wind-power events.